Evolution of mesoscopic interactions and scattering solutions of the Boltzmann equation

TL;DR

This paper develops a new theoretical framework to analyze the evolution and scattering solutions of particles governed by the Boltzmann equation on a mesoscopic scale, introducing novel bounds and demonstrating universal dispersion.

Contribution

It introduces an uncertainty principle and a priori bounds for mesoscopic particle interactions, and establishes a general scattering theory with existence and uniqueness results for Boltzmann solutions.

Findings

Evidence for universal dispersion of particles

Existence and uniqueness of scattering solutions

Asymptotic stability and completeness in $L^{inity}$

Abstract

The purpose of this paper is to study the evolution of moving interacting particles on the mesoscopic scale. We will introduce an uncertainty principle and a new priori bound for the evolution of particles subject to a general mesoscopic interaction and use this setting to demonstrate evidence for the universal dispersion of particles in the absence of the conventional methods like velocity averaging lemmas or Strichartz-type estimates. We will develop a scattering theory in generality and use its frame work to show the existence and uniqueness of a class scattering solutions to the Boltzmann equation with asymptotic stability and completeness in the $L^{\infty}$ setting.